galerkin

The Galerkin method is a numerical technique for solving differential equations by projecting the problem onto a finite-dimensional subspace. Named after the Russian mathematician B. C. Galerkin, it provides a systematic way to obtain approximate solutions that satisfy the equation in a weighted residual sense. In practice, one selects a finite set of basis functions and expresses the approximate solution as a linear combination of these functions.

Formally, let V be a function space suitable for the problem, and choose a finite-dimensional subspace V_h with basis functions {φ_i}. The Galerkin approach seeks u_h in V_h such that the residual is orthogonal to V_h: for each basis function φ_j, the inner product (R(u_h), φ_j) = 0, where R is the differential operator applied to u_h minus any source terms. This yields a system of algebraic equations for the coefficients in u_h. If the test space coincides with the trial space, the method is called the Galerkin method; if the test space differs, it is a Petrov–Galerkin method.

For partial differential equations, the process often involves a weak formulation: multiply the equation by test functions, integrate by parts to reduce differentiability requirements, and impose the Galerkin condition. An example is the Poisson equation -Δu = f, with boundary conditions, leading to a bilinear form a(u,v) and a linear form l(v); the approximate solution satisfies a(u_h,v) = l(v) for all v in V_h.

The Galerkin method is foundational to the finite element method. Convergence is ensured under stability and consistency conditions, with error controlled by the best approximation in V_h. Variants include spectral Galerkin and various Petrov–Galerkin formulations used in fluid dynamics and other fields.